J. Phys. Chem. 1996, 100, 15599-15604
15599
Thermodynamics of Ion Exchange. An Alternative Methodology Denver G. Hall NEWI, Plas Coch, Mold Road, Wrexham, LL11 2AW, U.K. ReceiVed: June 7, 1996X
The thermodynamics of ion exchange is usually presented in terms of ion exchange equilibrium constants. This formalism is somewhat cumbersome, especially when more than two ionic species are involved in the exchange equilibria. It is shown that an alternative formulation using a notation developed previously by the author simplifies matters. Clausius-Clapeyron equations are derived for enthalpies of exchange under the practically convenient condition of constant solvent activity rather than constant solvent content of the exchanger phase. A definition is forwarded of thermodynamically ideal behavior in the exchanger phase which corresponds as closely as possible with the thermodynamic terms in the Nernst-Planck equations describing ideal ion transport behavior. When Kirkwood-Buff solution theory is applied to fixed charge exchangers, it turns out that all terms explicitly involving the fixed charge distribution are zero.
Introduction Many papers have been written about the thermodynamics of ion exchange between electrolyte solutions and materials containing fixed charges such as zeolites, clays, and ionexchange resins. Much of this work has been concerned with exchange equilibria involving only two ionic species. However, ion exchange phenomena involving three or more ionic species have also received attention,1-3 and there has been some debate about issues such as the optimum choice of concentration scales in the exchanger phase,4 standard states, and the role of water activities. A feature of this area is that the customary notation is complex. (See, for example, the definition of thermodynamic equilibrium constants in refs 1-3.) A result of this is that it is difficult to display clearly some of the key issues. The aim of this contribution is to forward an alternative notation and procedure which lead to a concise and general formalism for the interpretation of experimental data in systems containing any number of exchangeable ionic species. An important feature of this alternative approach is that it does not rely on a favored choice of concentration scales or on any particular definitions of equilibrium constants and selectivity coefficients and by so doing circumvents the difficulties raised by issues of this kind. Also it readily accommodates phenomena such as swelling of the exchanger phase, salt imbibition, and the influence of solvent activity on exchange equilibria and provides a general basis for discussing the enthalpy and volume changes associated with ion exchange. A definition of thermodynamically ideal behavior in the exchanger phase which is compatible with the NernstPlanck equations5 describing ideal ion transport behavior is forwarded. Finally, the application of Kirkwood-Buff solution theory6,7 to ion exchange materials with mobile or fixed exchange sites is discussed briefly.
Ni moles each of a series of ionic species (i ) 1 f c). We write for this system
µw µˇ i µˇ z G V H d ) - dT + dp + dNz + ∑ dNi + dNw (1) T T T T i T T2 where most of the symbols have their usual meanings, G denotes Gibbs free energy, H denotes enthalpy, µˇ denotes electrochemical potential, etc. For bulk ion exchanger the electrical neutrality condition ensures that
∑i νiNi + νzNz ) 0
where V denotes ionic valence and includes the sign. Consequently the Ni and Nz cannot all be varied independently. Let λi be given by
(
λi ) µˇ i -
X
Abstract published in AdVance ACS Abstracts, August 15, 1996.
S0022-3654(96)01693-0 CCC: $12.00
)
νi µˇ νz z
(3)
For bulk exchanger eq 1 now becomes
µw λi V H G dNw d ) - dT + dp + ∑ dNi + T T T i T T2
(4)
The λi refer to electrically neutral entities and are well defined as appropriate derivatives of G. We need not concern ourselves with the precise definition of the µˇ i which is not so straightforward. Since the bulk exchanger phase is physically homogeneous
G ) ∑ λiNi + µwNw
Derivation of Fundamental Equations For simplicity it is supposed that the ion exchanger has only one kind of charged site. From a general thermodynamic point of view, the extension to several kinds of sites is quite straightforward but for systems at equilibrium is merely an extra complication. We consider an amount of exchanger with Nz moles of charged sites, and let the charge per mole of these sites be νzF. We let the system contain Nw moles of water and
(2)
(5)
i
and it follows from eqs 4 and 5 that
0)-
H T2
dT +
λi µw dp - ∑ Ni d - Nw d T T T i
V
(6)
which is the Gibbs-Duhem equation for the ion exchanger plus ions plus water. Dividing eq 6 by Nz we obtain for 1 mol of exchanger © 1996 American Chemical Society
15600 J. Phys. Chem., Vol. 100, No. 38, 1996
0)-
h T2
dT +
Hall
λi µw dp - ∑mi d - mw d T T T i V
(7)
where h ) H/Nz, V ) V/Nz, mi ) Ni/Nz, and mw ) Nw/Nz Let g ) G/Nz. Evidently
µw λi V h g dmw d ) - dT + dp + ∑ dmi + T T T i T T2
(8)
However, it follows from eq 2 that
∑i νi dmi ) 0
c-1θ µw i V h g dmw d ) - dT + dp + ∑ dmi + T T T i)1 T T2
) (
)
νi νi where θi ) λi - λc ) µˇ i - µˇ c νc νc
(11)
c-1m ν i i i)1
-
νc
νz νc
(λIIk - λIk) ) (θIIk - θIk) +
νk II (λ - λIc) νc c
(17)
When all ionic species in the bulk solution are present in the exchanger phase, the solution Gibbs-Duhem equation may be used to eliminate dµw from eqs 7 and 13. However, this procedure is less useful when there are species in solution which are not present in the exchanger phase. A drawback of eqs 16 and 17 is the special status accorded to species c. However, if θk and the θi are expressed in terms of chemical potentials in solution by writing
(12) θk ) µsk -
νk s µ νc c
we find that eq 7 may be rewritten as
θi µw Vz λc h V d ) dT - dp + ∑mi d + mw d Vc T T2 T T T i)1
θi )
c-1
(13)
where use has been made of eqs 11 and 12. Equation 7 has the advantage that all ionic species in the exchanger phase are treated in the same way. It is identical in form to the Gibbs-Duhem equation for a c + 1 component solution, and it is clear that the λi play the same roles as the chemical potentials in such a solution.
Consider now the situation in which the ion exchanger is in equilibrium with an electrolyte solution. In general, for fixed charge exchangers, this solution will contain at least all mobile species present in the exchanger and maybe other species as well. Let superscript s denote the solution phase. The conditions for equilibrium between the two phases are given by
µˇ is
(14a,b)
) µˇ i
(18a,b)
νi - µsc νc
then on substituting for λIIc - λIc as given by eq 17 into eq 17 and utilizing eqs 18a,b it follows straightforwardly that for any species k including c
λIIk ) λIk +
νk νz
[ (
νi
∫III ∑mi d µis - ν i
λk -
k
)
]
µsk + mw dµw
(19)
(15)
It is to be stressed that the θsi are well-defined quantities which refer to electrically neutral combinations of ions. Changes in the θsi and µw at constant T and p are measurable
νk νk λj ) µsk - µjs νj νj
(20)
which applies under all conditions. We may therefore use eq 20 to substitute for λIIk in eq 19. After some rearrangement followed by interchanging subscripts j and k, we obtain
λIIk )
νk νj
[
λjI + µsII k Vk Vz
from whence follows
θis ) θi
µis
It also follows from eq 11 that for any pair of species k and j
Equilibrium with Electrolyte Solutions
µws ) µw
(16)
and is independent of the path of integration. Changes in the other λk are given by
(10)
The choice of species c is arbitrary and can be made in any way that is convenient. Bearing in mind that
mc ) - ∑
c-1 νz II (λIIc - λIc) ) ∫I (∑mi dθi + mw dµw) νc i)1
(9)
Hence, not all the mi in eq 8 are independently variable. We choose mi (i ) 1 f (c - 1)) as the independent set and rewrite eq 8 as
(
experimentally using classical techniques such as EMF measurement and the determination of vapor pressures. Changes in the θsi can also be calculated if accurate estimates of mean ionic activity coefficients are available. It is apparent from the above that studies of ion exchange equilibria enable the mi and mw to be determined as functions of the θi and µw. However, it is the dependence of the λi on the mi and mw that is the goal of experimental investigations. The relationship between the λi and the θi is readily obtained from eq 13. This equation gives on integration between two states I and II
νk νj
]
µjsII +
[ (
νi
∫III ∑mi d µis - ν µjs i
j
)
]
+ mw dµw (21)
The choice of species j in eq 21 can be made at will. We note that the single-ion chemical potentials in eqs 19 and 21 always occur in electrically neutral combinations which are well defined thermodynamically. In these equations I and II refer to any two states. Hence state II may be taken as an
Thermodynamics of Ion Exchange
J. Phys. Chem., Vol. 100, No. 38, 1996 15601
arbitrary state and state I taken as a reference state. The specification of an appropriate reference state is also to some extent arbitrary. When k is a counterion, the reference state may be taken as a homoionic k form of the exchanger in which µw has some specified value. Taking µw as the chemical potential of pure water at the T and p of interest is an attractive option, and in this case λIk depends only on T and p. Alternative choices of µw may sometimes be more convenient. By applying eq 19 it is straightforward to obtain λIIk in terms of λIk and the composition of the exchanger phase from appropriate measurements of the ion exchange equilibria. When k is a coion or a counterion whose homonionic form is inaccessible, an appropriate reference state is one in which mIk is zero. Since λIk and the integrand on the right-hand side of eq 19 are ill defined in this limit, it is preferable to use eq 21. Any state in which mk is zero may be taken as state I, and it may often be convenient to choose this state as a homoionic form as outlined above. In this case j should be chosen as the counterion of the homoionic form concerned. If λIj in eq 21 does not refer to a homoionic form, eq 19 can be used to express it in terms of λj for the homoionic j form and an appropriate integral. Equations 19 and 21 have the important advantage that they do not rely on any particular choice of concentration units in the exchanger phase or on any particular definitions of selectivity coefficients and equilibrium constants. Partial Molar Quantities Partial molar quantities may be defined for the exchanger phase in just the same way as for a multicomponent solution. For example at constant T and p we may write
() ()
λi ∂H T hi ) ) ∂Ni T,p,Nj,Nw ∂T
( )
∂
λi T ) p,Ni,Nw ∂T
constant T, p, and µw, we have
H ) ∑ hi* Ni
(27)
i
Similar reasoning also applies to Nw. The quantities mi* may be defined by writing
m i* )
( ) ∂Nw ∂Ni
)T,p,Nj,µw
( ) ∂λi ∂µw
T,p,Ni
(29)
i
It is apparent from eq 28 that mi* dNi is the amount of water which must be added to the exchanger phase to keep µw constant when Ni is increased by dNi keeping the other Nj constant and describes the effect of water activity on λi at constant T, p, and mi. Often mi* might be expected to be much the same as Nw/ Ni for the homoionic i form of the exchanger and not to depend greatly on the other Nj. However this may not be the case when appreciable swelling can occur. The hi* may be related to the hi as follows. We have
( ) ∂H ∂Ni
)
T,p,Nj,µw
( ) ∂H ∂Ni
+
T,p,Nj,Nw
( ) ( ) ∂H ∂Nw
p,mi,mw
T,p,Ni
() () λi T ∂T
(23)
∂
λi T ) p,mi,mw ∂T ∂
( )
∂λi + p,mi,µw ∂µw
hw is defined in the same way as the hi and
dH ) ∑ hi dNi + hw dNw
(24)
i
Partial molar volumes can be introduced in the same way. Since the procedures are well established, they will not be pursued further here. Instead an alternative approach will be examined. The principle underyling this alternative approach is that in practice the quantity nw is less easy to control than µw. For this and other reasons µw may be a preferable variable to work with. At constant T, p, and µw, H ) H(Ni) and we have
dH ) ∑ hi* dNi
p,Ni,(µw/T)
(26)
Also since H is a linear homogeneous function of the Ni at
() µw T ∂T
∂
T,p,mi
p,mi,mw
(32)
For the solution phase θsk may be regarded straightforwardly as a function of T, p, and solution composition so that
θsk
d
T
hsk )-
νk s h νc c
T2
θsk T
)
1 T
νk s V νc c θsk dp + D T T
Vsk dT +
where
D
()
(31)
Clausius-Clapeyron Equations
(25)
i
()
(30) T,p,Nj,µw
hi* ) hi + mi*hw
(22)
i
( )
∂Nw ∂Ni
which is of course the same as
H ) ∑ Nihi + Nwhw
λi ∂ T ) p,Ni,(µw/T) ∂T
(28) T,p,mi
Nw ) ∑ miNi
Equation 31 is also equivalent to
λi ∂ ∂H T ) h i* ) ∂Ni T,p,Nj,µw ∂T
∂λi ∂µw
and noting that
∂
where
where
( )
)-
∑i
() ∂θsk
∂mis T,p,msj
dmis
(33)
(34)
and the msi are a set of independent composition variables for the solution, typically molalities or mole ratios. Quantities such as hsk and Vsk in eq 33 refer to individual ionic species but occur in electrically neutral combinations. Regarding θk/T in the exchanger phase as a function of T, p, and the mi, and µw/T enables us to write
15602 J. Phys. Chem., Vol. 100, No. 38, 1996
θk d
hk* )-
T
νk νc
Vk* -
hc* dT +
T2
νc
T
() ∂θk
1
νk
∑ T i ∂m
( ) ∂V ∂µw
Vc*
(
dp +
dmi - mk* -
i T,p,mj,µw
Hall
)
νk νc
µw
mc* d
T
(35)
for variations at equilibrium dθi/T ) dθsi /T and dµwT ) dµws/T where
Vws hws µws µws d ) - 2 dT + dp + D T T T T
(36)
and Dµws / T is defined in the same way as Dθsk / T in eq 34. We now substitute eq 36 into eq 35 and equate the right-hand side of the resultant expression with that of eq 33. At constant mi we obtain
0)
∆hck T2
dT -
(
)
∆Vck νk µws θsk dp + D + mk* - mc* D (37) T T νc T
T,p,mi
( (
) ( ) (
) ( ) (
) )
νk νk νk hc* - hsk - hsc - mk* - mc* hws νc νc νc
(38a,b)
νk νk νk ) Vk* - Vc* - Vsk - Vsc - mk* - mc* Vws νc νc νc
The quantities on the left-hand side of eqs 38a,b refer to the process whereby 1 mol of k is transferred from the solution to the ion exchanger and an equivalent amount of c is transferred either from the exchanger to the solution when k and c have the same sign or from the solution to the exchanger when k and c have opposite signs. The transfer takes place under the condition that the solvent is allowed to equilibrate but all other mj are held constant. For binary exchange ∆hck is clearly related to the enthalpy of exchange obtainable from calorimetric experiments. Equation 37 provides a relationship between T, p, and the bulk solution composition for which the mi in the exchanger phase are held constant. If there are c ionic species in the exchanger then there are c - 1 independent equations of this kind. (hw - hws) is the enthalpy of transfer of water from solution to the exchanger phase and is determinable from the temperature dependence at constant composition of the exchanger phase and solution phase vapor pressures. Hence enthalpies of exchange at constant solvent content can be found in principle from the enthalpies at constant solvent activity discussed above.
) T,mi,µw
( ) ∂mw ∂µw
Vw
For the case considered above, namely an isotropic exchanger phase, eq 10 may be rearranged to give
)
g - mwµw T
Thermodynamics plays an important role in understanding both the equilibrium behavior of ion exchangers and their transport properties. The definition of ideal behavior is to some extent arbitrary, and there has been some debate about whether or not ion fractions or charge fractions are the most appropriate concentration units to use.4 Reasons have been forwarded for preferring the latter.4 The discussion of ion transport phenomena is often based on the Nernst-Planck equations5 which may be regarded as defining ideal transport behavior. Implicit in these equations is the assumption that gradients in electrochemical potentials are given by
(41)
where grad φ denotes the electric field strength and RT grad ln ni is the chemical potential gradient where n denotes amount per unit volume. To ensure that the definitions of ideal transport behavior and ideal thermodynamic behavior correspond as closely as possible, it is proposed that the latter be defined as that whereby the chemical potentials of the individual ionic species in the exchanger phase are given by
µi ) µiθ (T, p, µw) + RT ln mi
(42)
This equation is compatible with eq 41 when gradients in µw and in V the volume per mole of exchange sites are both negligible. According to eq 42
θi ) θiθ (T, p, µw) + RT ln mi -
νi RT ln mc νc
(43)
which gives together with eq 13 at constant T, p, and µw
νz νc
[
dλc ) RT ∑ mi d ln mi i*c
νi νc
]
mc d ln mc
(44)
On integration between two states I and II eq 44 gives
λIIc
-
λIc
νc mIIc II I ) RT [∑(mi - mi )] + RT ln νz i mI
(45)
c
which together with eq 17 gives
λIIk
)
(40)
T,p,mi
Ideal Behavior in the Exchanger Phase
Swelling of the Exchanger Phase
(
∂mw ∂p
grad µˇ i ) RT grad ln ni + νiF grad φ
∆hck ) hk* -
d
( )
where Vw is the partial molar volume of water in the exchanger phase. Equation 40 shows that swelling effects are related to the effect of pressure on the water content at constant mi and µw. For systems such as clays which are not isotropic the conditions for mechnical equilibrium are more complex.8
where
∆Vck
)-
-
λIk
νk mIIk II I ) RT [∑(mi - mi )] + RT ln νz i mI
(46)
k
-
h 2
T
dT +
c-1θ µw i dp + ∑ dmi - mw d (39) T T i)1 T
V
which on cross differentiation enables us to write
Equation 46 is valid for counterions and coions alike. For counterions it is appropriate to take state I as the homoionic form of k at the T, p, and µw of interest in which case it is straightforward to show that
Thermodynamics of Ion Exchange
λk ) λ0k (T, p, µw) + RT ln
mkνk
∑miνi
J. Phys. Chem., Vol. 100, No. 38, 1996 15603
+ RT
∑i mi(νi - νk) ∑miνi
i
(47)
i
The corresponding result for coions can be shown to be
λk )
λθk
(T, p, µw) + RT ln mk + RT
∑i mi(νi - νk) ∑i miνi
(48)
In deriving these expressions use has been made of eq 12. Equations 47 and 48 confirm the notion that charge fractions are preferable to ion fractions as concentration units in accordance with the treatment of Gaines and Thomas.9 It is notable that eq 47 is exactly analogous to the equations of polymer solution thermodynamics when the so-called χ parameter is zero with the charge fractions playing the same role as partial molar volumes. Moreover, if the final terms of eqs 47 and 48 are omitted, the resultant expressions are no longer consistent with the Gibbs-Duhem equation for the exchanger phase (eq 7). It is of course more or less self-evident that the NernstPlanck equations cannot be expected to apply when eqs 47 and 48 do not describe adequately the behavior of the λi. Activity coefficients in the exchanger phase may be defined in the usual way in terms of λk - λk(ideal) where λk(ideal) is given by eq 47 or 48. We will not elaborate further on this issue because to do so would add nothing new of any physical import. Application of Kirkwood-Buff Solution Theory For ordinary solutions the Kirkwood-Buff theory6,7 provides general and exact relationships between pair distribution functions and the dependence of chemical potentials on solution composition. According to this theory, we may write at constant T for any solution species β
kT d ln nβ ) ∑ (NβR + δβR)dµˇ R
(49)
where n denotes number density, the summation over R includes all species in solution, δβR denotes the Kronecker δ ()0 when β * R, )1 when β ) R), NβR is defined by ∞
(50)
and gβR(r) is the pair distribution function of R with respect to β. Evidently
nβNβR ) nRNRβ
(51)
Also, in electrolyte solutions, the condition that the charge of an ion is balanced by that of its atmosphere ensures that
∑R (NβR + δβR)νR ) 0
(52)
Consequently eq 49 may be written as
kT d ln nβ )
∑ (NβR + δβR)dλR
R*z
0 ) ∑ Nzi dµˇ i + Nzw dµw ) ∑ Nzi dθi + Nzw dµw i
(54)
i*c
Since the θi and µw are independent variables, it follows from eq 53 that
Nzi ) Nzw ) 0
(55)
which together with eq 51 give
Niz ) Nwz ) 0
(56)
It is concluded therefore that in this case the thermodynamic behavior of the exchanger phase can be described entirely in terms of the distribution functions of the mobile ionic species and water. Equations 53, 54, and 55 enable us to write
kT d ln nk ) ∑ (Nki + δki)dθi
(57)
i*c
R
NβR ) nR ∫0 (gβR(r) - 1)4πR2 dr
The theory may be applied as it stands to liquid ion exchangers in which the exchanger sites are mobile. Let 0 and 1, respectively, denote the solvent base of the exchanger and the mobile exchanger sites. In this case there is an equation analogous to eq 49 for all species in the exchanger phase. The Gibbs-Duhem equation for the exchanger phase may be used to eliminate dµ0 from these equations thus enabling the quantities kT d ln(ni/n0) and kT d ln(nz/n0) to be expressed in terms of the dλi and dµw or, alternatively, in terms of the dθi, dλc, and dµw. Moreover when species 0 and z are completely absent from the solution in equilibrium with the exchanger, it is clear that for a given exchanger kT d ln(nz/n0) ) 0. This provides an expression relating dλc to the dθi and dµw which may now be used to express any kT d ln mi in terms of the dθi and dµw. It turns out that the coefficient of dθk in the expression for kT d ln mk approaches 1 as mk f 0 provided that in this limit the k ions are randomly distributed. Ion exchangers with fixed sites raise the interesting issue of how these sites should be handled. An extreme sample of this situation is that where the exchanger phase is incompressible, does not swell and the fixed sites are arranged on a lattice. In this case nz is constant and Nzz ) -1 so that eq 49 gives
(53)
where for ionic species λR is defined by eq 3 and the λ values for uncharged species are their chemical potentials.
For the homoionic form of the exchanger, since dhk ) 0, it follows from eq 57 that Nkk ) -1. When we have ideal thermodynamic behavior as described above, the Nki for the various mobile ionic species present are given by
Nki ) -
n i νi νk
∑njνj2
(58)
j
where the summation does not include the fixed charges. A similar equation to eq 58 applies to liquid ion exchangers, but in this the mobile sites do appear in the summation. A consequence of eqs 55 and 56 is that ion binding to the fixed site does not necessarily entail nonideal behavior as defined above. In general, however, it seems unlikely that the NernstPlanck equations will apply when a significant fraction of the fixed charges are involved in site binding. For mobile and fixed site exchangers alike the above theory may be used to show that if species k is randomly distributed in the exchanger phase as mk f 0 then in this limit
θk ) θθk (T, p, θi(i*k), µw) + kT ln mk
(59)
Together with eq 13, eq 59 enables a similar expression to be
15604 J. Phys. Chem., Vol. 100, No. 38, 1996 derived for λk. Equation 50 provides a firm basis for expressions which describe the limiting behavior of coion inbibition, in particular, when the bathing solution and the exchanger have but two ionic species in common. Real ion exchange resins presumably fall between the two extremes of completely mobile sites and fixed sites on a rigid lattice. In many cases they probably approximate more closely with the latter extreme than the former as arguments given elsewhere10 indicate. Concluding Remarks For binary exchange at constant T and p the above treatment is essentially equivalent to that presented by Gaines and Thomas9 over 40 years ago. In particular the λi defined above correspond with their chemical potentials. Whereas Gaines and Thomas couched their work in terms of exchange equilibrium constants, the present paper uses instead a notation applied previously by the author in a variety of situations.11 A major feature of the present work is that the solvent chemical potential is treated as an environmental variable in the same way as temperature and pressure. The formalism which results is particularly convenient for discussing the effects of temperature and pressure on exchange equilibria and also when the exchanger contains more than two exchangeable species. It has been used to derive Clausius-Clapeyron equations for exchange enthalpies under the practically relevant condition of constant solvent activity which is obeyed closely by most systems of practical interest. The definition of thermodynamically ideal behavior in the exchanger forwarded above has the important advantage that it is more compatible with the Nernst-Planck
Hall equations used to describe transport phenomena in the exchanger than the alternative definition based on ion fractions as concentration units. The application of Kirkwood-Buff solution theory to ion exchangers has been confined deliberately to the discussion of quantities which are thermodynamically respectable insofar as they are concerned only with the measurable properties of electrically neutral combinations of ions. A more extended account which involves the thermodynamic properties of individual ionic species is presented elsewhere.10 References and Notes (1) Fletcher, P.; Townsend, R. P. J. Chem. Soc., Faraday Trans. 2 1981, 77, 955; 965, 2077; 1983, 79, 419. (2) Fletcher, P.; Franklin, K. R.; Townsend, R. P. Philos. Trans. R. Soc. London A 1984, 312, 141. (3) Franklin, K. R.; Townsend, R. P. J. Chem. Soc., Faraday TRans. 1 1985, 81, 1071. (4) Barrer, R. M.; Townsend, R. P. J. Chem. Soc., Faraday Trans. 2 1984, 80, 629. Barrer, R. M.; Townsend, R. P. Zeolites 1985, 5, 287. (5) Morf, W. E. The Principles of Ion-SelectiVe Electrodes and of Membrane Transport, Studies in Analytical Chemistry; Elsevier: Amsterdam, 1981; Vol. 2. (6) Kirkwood, J. G.; Buff, F. P. J. Chem. Phys. 1951, 19, 774. (7) Hall, D. G. Trans. Faraday Soc. 1971, 67, 2516. (8) Gibbs, J. W. The Scientific Papers of JW Gibbs; Longmans: London, New York, 1928; Vol. 1, pp 184-218. (9) Gaines, G. L., Jr.; Thomas, H. C. J. Chem. Phys. 1953, 21, 714. (10) Hall, D. G. Proceedings of the 4th International Conference and Industrial Exhibition on Ion Exchange Processes ION-EX ‘95. To be published. (11) Hall, D. G. J. Chem. Soc., Faraday Trans. 2 1972, 68, 1439, 2169; 1973, 69, 1391. Hall, D. G. J. Chem. Soc., Faraday Trans. 1 1978, 74, 405; 1981, 77, 1121; 1985, 81, 885.
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